A Theorem on Subdivision Approximation and Its

Application to Obtaining PTAS’s for Geometric Optimization

Problems

Joseph S.B. MitchellStony Brook University

Workshop on Computational and Conformal Geometry, April 21, 2007

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A DedicationA Dedication

Happy 60th Birthday!!

Michael Ian Shamos: PhD thesis “Computational Geometry”, Yale University, 1978

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Motivating Problem: TSP with Motivating Problem: TSP with NeighborhoodsNeighborhoods

Find shortest tour to visit a set of neighborhoods P1,P2,…,Pn

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TSPN for Disk PackingTSPN for Disk Packing

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TSPN in a Circle PackingTSPN in a Circle Packing

"Introduction to Circle Packing: the Theory of Discrete Analytic functions"by Ken Stephenson

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Another (Springtime) Another (Springtime) MotivationMotivation

Best method of mowing the lawn?

TSPN: Visit the disk centered at each blade of grass

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Watchman Route ProblemWatchman Route Problem

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Sensor Network Application: Sensor Network Application: Cover Tour ProblemCover Tour Problem

Alt, Arkin, Bronnimann, Erickson, Fekete, Knauer, Lenchner, Mitchell,Whittlesey, SoCG’06

Min: Tour length + C * (sum of radii)

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Sensor Network Application: Sensor Network Application: Minimizing # Relay StationsMinimizing # Relay Stations

Efrat, Fekete, Mitchell 2007

Goal: Connect all subnetworks of sensors using min # of new relay stations

New result: PTAS

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Min-Weight Convex SubdivisionMin-Weight Convex Subdivision

Steiner version

Special Case: Min-weight (Steiner) triangulation

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Approximation AlgorithmsApproximation Algorithms

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Background on TSPBackground on TSP

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PTAS for Geometric TSPPTAS for Geometric TSP

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TSPN Recent ResultTSPN Recent Result

TSPN has a PTAS for TSPN has a PTAS for regions/neighborhoods that are regions/neighborhoods that are ““fatfat”, disjoint (or sufficiently disjoint) ”, disjoint (or sufficiently disjoint) connected regions in the planeconnected regions in the plane

Applies also to “MST with Applies also to “MST with neighborhoods”, Steiner MSTN, and neighborhoods”, Steiner MSTN, and many related problemsmany related problems

PTAS = Polynomial-Time Approximation Scheme = (1+)-approx, any >0

[SODA’07]

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Background on TSPNBackground on TSPN

Generalizes 2D Euclidean TSP (thus, NP-Generalizes 2D Euclidean TSP (thus, NP-hard)hard)

Introduced by Introduced by [Arkin & Hassin, 1994][Arkin & Hassin, 1994]• ““obvious” heuristics do not work:obvious” heuristics do not work:

TSP approx on centroids (TSP approx on centroids (as representative pointsas representative points)) Greedy algorithms (Greedy algorithms (Prim- or Kruskal-likePrim- or Kruskal-like) )

• O(1)-approx, time O(n + k log k), for “nice” O(1)-approx, time O(n + k log k), for “nice” regions:regions:

Parallel unit segmentsParallel unit segments Unit disksUnit disks Translates of a polygon PTranslates of a polygon P

• Combination LemmaCombination Lemma

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General Connected RegionsGeneral Connected Regions

O(log k)-approxO(log k)-approx [Mata & M, SoCG’95][Mata & M, SoCG’95]

Use guillotine rectangular subdivisions, Use guillotine rectangular subdivisions, DPDP

((nonnon – disjoint: regions may overlap) – disjoint: regions may overlap)

O(nO(n55) time) time

[Mata & M, SoCG’95][Mata & M, SoCG’95]

O(nO(n22 log n) log n)[Gudmundsson & Levcopoulos, 1999][Gudmundsson & Levcopoulos, 1999]k = # regions n = # vertices of all regions

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O(1)-ApproximationsO(1)-Approximations Unit disks, parallel unit segments, translates of Unit disks, parallel unit segments, translates of PP

[Arkin & Hassin, 1993][Arkin & Hassin, 1993]

Connected regions of comparable sizeConnected regions of comparable size [Dumitrescu & M, SODA’01][Dumitrescu & M, SODA’01]

Disjoint Disjoint fatfat regions of regions of anyany size size [de Berg, [de Berg, Gudmundsson, Gudmundsson, Katz, Levcopoulos, Overmars, van Katz, Levcopoulos, Overmars, van der Stappen, ESA’02]der Stappen, ESA’02]

Discrete point sets within disjoint, fat, Discrete point sets within disjoint, fat, nonnon-convex -convex regionsregions

[Elbassioni, Fishkin, Mustafa, Sitters, [Elbassioni, Fishkin, Mustafa, Sitters, ICALP’05]ICALP’05]

Non Non - disjoint, convex, fat, comparable size- disjoint, convex, fat, comparable size [Elbassioni, Fishkin, Sitters, [Elbassioni, Fishkin, Sitters, ISAAC’06]ISAAC’06]

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PTAS: O(1+PTAS: O(1+)-)-ApproximationsApproximations

Disjoint (or nearly disjoint) Disjoint (or nearly disjoint) fatfat regions of comparable regions of comparable size size [Dumitrescu & M, SODA’01][Dumitrescu & M, SODA’01]

Point clusters within disjoint Point clusters within disjoint fatfat regions of regions of comparable size in Rcomparable size in Rdd

[Feremans, Grigoriev, [Feremans, Grigoriev, EWCG’05]EWCG’05]

NewNew: PTAS for disjoint (or nearly disjoint) : PTAS for disjoint (or nearly disjoint) fatfat regions of regions of arbitraryarbitrary sizes. sizes.

Def: P is fat if area( P ) = ( diam2(P) )Weaker notion than usual “fatness”

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Related Work: APX-hardnessRelated Work: APX-hardness General connected regions (overlapping):General connected regions (overlapping):

• No c-approx with c<391/390, unless P=NPNo c-approx with c<391/390, unless P=NP

[de Berg, Gudmundsson, Katz, [de Berg, Gudmundsson, Katz,

Levcopoulos, Levcopoulos, Overmars, van der Stappen, Overmars, van der Stappen,

ESA’02]ESA’02]

(from MinVertexCover)(from MinVertexCover)

• No c-approx with c<2, unless P No c-approx with c<2, unless P j TIME (n TIME (nO(log log n)O(log log n)))

[Safra, Schwartz, [Safra, Schwartz,

ESA’03]ESA’03](from Hypergraph VertexCover)(from Hypergraph VertexCover)

Line segments, comparable lengthLine segments, comparable length[Elbassioni, Fishkin, Sitters, ISAAC’06][Elbassioni, Fishkin, Sitters, ISAAC’06]

Pairs of points (disconnected)Pairs of points (disconnected) [Dror, Orlin, [Dror, Orlin, 2004]2004]

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Exact Poly-Time SolutionsExact Poly-Time Solutions

TSPN for a set of infinite lines in 2D:

Solved in O(n4 log n) time using Watchman Route solution [Dror, Efrat, Lubiw, M, STOC’03]

Is this the only nontrivial case exactly solvable in poly-time?

What about visiting planes in 3D?

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Recipe for PTASRecipe for PTAS

OPT Network with specialrecursive structure

increasing length by (1+) factor

Use dynamic programming to compute shortest network with the required structure (connectivity, Eulerian subgraph, etc)

Optimal network with special structure TSPN tour

Structure Theorem

What should the special recursive structure be?

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m-Guillotine Structurem-Guillotine Structure

Network edge set E is m-guillotine if it can be recursively partitioned by horiz/vertical cuts, each having small (O(m)) complexity wrt E

Example: 3-guillotineEach cut intersects E in at most 3 connected components

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Desired Recursive StructureDesired Recursive Structure

Constant (O(m))

information flow across boundary

Rectangular subproblem in dynamic program (recursion)

cut

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m-Guillotine Structure Theoremm-Guillotine Structure Theorem

Any set E of edges of length L can be made to be m-guillotine by adding length O(L/m) to E, for any positive integer m.

Proof is based on a simple charging scheme.

While this “scribble” may not be m-guillotine, it is “close”

in that it can be made m-guillotineby adding only (1/m)th of its length

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Possible Vertical CutsPossible Vertical Cuts

f(x)

xf(x) = length of m-bridge

= cost of construction

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Paying for the Bridge Paying for the Bridge Construction: The Chargeable Construction: The Chargeable

LengthLength

h(x) = chargeable length

xGreen portion: “m-dark”

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Charging SchemeCharging Scheme

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Charging SchemeCharging Scheme

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Charging SchemeCharging Scheme

x

y

f(x) = cost of construction of vert cut at x

h(y) = chargeable length of horiz cut at y

Red area = =

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Charging SchemeCharging Scheme

y

Blue area =

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Charging SchemeCharging Scheme

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SubproblemSubproblem

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Dynamic Program: Dynamic Program: Min Steiner TreeMin Steiner Tree

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Difficulty in Applying TSP Difficulty in Applying TSP Methods to TSPN / MSTNMethods to TSPN / MSTN

Consider a subproblem (rectangle):

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New StructureNew Structure

Build Build region-bridgesregion-bridges in order to encode in order to encode succinctly which regions are the succinctly which regions are the “responsibility” of a subproblem“responsibility” of a subproblem

Cannot afford to build Cannot afford to build mm-region-bridges-region-bridgesfor for m = O(1/m = O(1/),), constant wrt constant wrt nn..

But But cancan afford to build afford to build M-M-region-bridges, region-bridges, with with M=O((1/M=O((1/)log n))log n) and this is “just and this is “just right”, since the remaining right”, since the remaining MM bridges that bridges that are not part of the bridge can be specified are not part of the bridge can be specified in the subproblem: in the subproblem: 22MM = = 22O(log n)O(log n) is is poly(n)poly(n)

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Subproblem: A Window into Subproblem: A Window into OPTOPT

Region-Bridges

M=3

Bridgesm = 4

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Subproblem OptimizationSubproblem Optimization

Region-Bridges

Bridges

Specification of a subproblem:

1. Window W

2. 4 Bridges, 2m segs/side of W

3. 4 Region-Bridges, one bit per 8M non-bridged crossing region: Is the subproblem responsible to visit?

4. Protruding regions not in RW0

specified by 2 sequences per side

5. Connection pattern among the O(m) segs crossing into W m = 1/

M = (1/) log n

W

mO(m)

nO(1)

nO(1)+2O((1/)log n) = nO(m)

n4

mO(m)

Total # subproblems = nO(m)

and nO(m) choices for the best horiz/vertical cut, in DP optimization

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(m,M)-Guillotine Structure(m,M)-Guillotine Structure

Definition: Network edge set E is (m,M)-guillotine if it can be recursively partitioned by horiz/vertical cuts, each containing the “m-span” (bridge) of E and the “M-region-span” (region-bridge) of the set of regions.

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What the DP ComputesWhat the DP Computes

Minimum-length network that is

1. (m,M)-guillotine wrt window W0 and regions RW0

2. Connected

3. Containing an Eulerian spanning subnetwork

4. Spanning (visits all regions)

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Main Idea of PTASMain Idea of PTAS

Use m-guillotine PTAS method, with new structure to address difficulty with TSPN

OPT (m,M)-guillotine networkwith special structure

increasing length by (1+) factor

Use dynamic programming to compute shortest (m,M)-guillotine

network with the required structure (connectivity, Eulerian subgraph, etc)

Optimal (m,M)-guillotine network with structure TSPN tour

Structure Theorem

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New Structure TheoremNew Structure Theorem

Theorem: Let E be a connected set of edges of length L, spanning all regions. Then, for any positive integers m and M, there is a superset, E’, of E, of length at most L + (sqrt(2)/m) L + (sqrt(2)/M) (RW0

).

Key Lemma: L* C (RW0) / log n

We pick M = (1/) log n , and m = 1/

Then, by the Key Lemma, we see that T* can be converted to be (m,M)-guillotine, adding length O(T*/)

Sum of region diameters

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The sum of the perimeters of a set of n disjoint fat regions that are visited by a path of length L is at most O(L log n)

Uses PACKING argument

Ex: Bound is tight

Key Geometric Observation

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Key Lemma: Lower Bound on Key Lemma: Lower Bound on OPTOPT

Key Lemma: L* C (RW0) / log n

Relates tour length of OPT, L*, to sum of diameters, (RW0)

Ex: Tight

L* = ( (RW0) / log

n)

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Proof of Key LemmaProof of Key Lemma

Let Ai = area( (T* + B(di )) W0 )Minkowski sum with ball of radius di

Cluster regions by size (diameter), into log (n/) classes

There are ni regions with diameter in range (di/2,di)

Area (packing) argument:

Claim: Ai 2di L*

By fatness, Ai C0 di2 ni , for some constant C0

Thus, by Claim below, C0 di2 ni 2di L*, or L* (C0/2) di

ni

Summing on i, we get L* C (RW0) / log n

This is where fatness and disjointness are

used!

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Bucketing Regions by SizeBucketing Regions by Size

Consider each minimal covering AAB, Consider each minimal covering AAB, WW00, snapped to grid, snapped to grid

RRWW00 = regions fully within = regions fully within WW00

PartitionPartition RRWW00 into into

K=O(log(D/K=O(log(D/))=O(log(n/))=O(log(n/)))) classes classes according to the diameters falling in according to the diameters falling in ranges:ranges:

(only O(n4) of them)

(each has diameter O(D))

(0,) (,2) (2,4) (4,8) … (2i-1,2i) … (2K-2,2K-1)

di=2i = D/n

Class i

Can shrink to single grid points, by Grid Lemma

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Proof of ClaimProof of Claim

Claim: Ai = area( (T* + B(di )) W0 ) 2di L*Proof:

area( T* + B(di ) ) 2di L* + di2

That portion within W0 does not include (at least) the area, di

2

W0=BB(T*)

Minkowski sum with ball of radius di

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Main ResultMain Result

Theorem: TSPN for disjoint fat regions has a PTAS.

PTAS also for the case of nondisjoint regions, if there are disjoint disks 1,…,n, with i Pi and diam(Pi)/diam(i) < C.Improve running time to O(nC), with C

independent of 1/: use grid-rounded guillotine subdivisions.

¿

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Generalizations/ExtensionsGeneralizations/Extensions

Disconnected regions: sets of Disconnected regions: sets of points/regions that are within a points/regions that are within a “nice” set of regions“nice” set of regions

k-TSPNk-TSPN Steiner MST with NeighborhoodsSteiner MST with Neighborhoods MST with Neighborhoods (MSTN)MST with Neighborhoods (MSTN) k-MSTNk-MSTN

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Approximation of 2D TSPN: Approximation of 2D TSPN: Connected RegionsConnected Regions Fat Regions non-Fat Regions

Comparablesizes

Arbitrarysize

Disjoint

Non-Disjoint

Disjoint

Non-Disjoint

Disjoint

Non-Disjoint

Disjoint

Non-Disjoint

PTAS

O(1)

O(1)

O(1)O(1), PTAS?

PTASNew

O(log n)

O(log n)

APX-hard

O(log n)

O(1)

APX-hard

Conjecture: PTAS for all

Conjecture: O(1) for all

PTASNewest

Newest

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Laundry List of ProblemsLaundry List of Problems

Know PTASKnow PTAS• TSP, k-TSP, Steiner TSP, k-TSP, Steiner

MST, k-MSTMST, k-MST• Red-blue separationRed-blue separation• Min-weight convex Min-weight convex

subdivisionsubdivision• TSPN, fat regionsTSPN, fat regions• Orienteering Orienteering

problemproblem• Lawnmowing Lawnmowing

problemproblem

OPEN: PTAS?OPEN: PTAS?• TSPN, disjoint regions in 2DTSPN, disjoint regions in 2D• Vehicle routing; min-weight Vehicle routing; min-weight

cover with k-tourscover with k-tours• Deg-3, deg-4 spanning Deg-3, deg-4 spanning

treestrees• Min-weight triangulationMin-weight triangulation• Watchman route problemWatchman route problem• Min-area triangulated Min-area triangulated

surface; special case: surface; special case: terrainterrain